Moments of the Derivative of Characteristic Polynomials with an Application to the Riemann Zeta Function |
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Authors: | J. B. Conrey M. O. Rubinstein N. C. Snaith |
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Affiliation: | (1) American Institute of Mathematics, 360 Portage Ave, Palo Alto, CA 94306, USA;(2) School of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom;(3) Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada |
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Abstract: | We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann ζ function on the critical line. We do the same for the analogue of Hardy’s Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum. |
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